VETTER: NATURAL MORTALITY IN FISH STOCKS 



C, =N,,i (FJF, + A/)(e' 



(2) 



If M is large relative to F (i.e., the exploitation 

 ratio is low), then errors in A^, can increase pro- 

 gressively and become quite large at the younger 

 ages (e.g., Agger et al. 1973; Murphy 1965; Ull- 

 tang 1977; Sims 1982a, 1982b, 1984). 



Numeric Results 



Although general responses of various models 

 can be determined simply by inspection of the 

 analytic models themselves, the quantitative 

 change to expect in the result (output) for a quan- 

 tified change in M (input) is not always immedi- 

 ately obvious. This is because M tends to occur 

 more than once in various formulas. For example, 

 M appears in both the numerator and denomina- 

 tor in the solution to the Beverton-Holt yield 

 equation (Ricker 1975). 



Y = FA^oe'-^'-WxdAM + F) 



-3e'-*'"V(M +F + k) 



+ 3e(-2*^V(M +F + 2k) 



e'-s^'-'/CM +F ^3k)). 



(3) 



So, rather than derive analytical expressions 

 (e.g., Sims 1984), I resort below to a simpler ap- 

 proach. Sensitivity of fishery models to changes of 

 given magnitude in M is assessed by comparing 

 percent change reported in model response (out- 

 put) to percent change in M (input). In cases for 



vector (age or density-dependent) M, I have 

 merely described the shape of the M -vector. For 

 these different vectors, I report the percent 

 change in the result due to switching from a vec- 

 tor of one shape to a vector of another shape. 



Yield Models 



At least four studies (Beverton and Holt 1957; 

 Hennemuth 1961; Francis 1974; Bartoo and Coan 

 1979) have shown that errors in estimates of 

 M propagate into roughly equal errors in esti- 

 mates of (y//? )maxj but with sign reversed (Table 

 2). For example, a 10% overestimate in M will 

 lead to approximately 10% underestimate of (Y/ 

 R 'max- An equally important result is that the 

 actual magnitude of the effect induced depends 

 strongly not just on the error in M, but on the 

 values chosen for the other parameters in the 

 model. 



In another study, Beddington and Cooke (1983) 

 used the Beverton-Hol formulation to investi- 

 gate the influence of M (constant; 0.1 to 0.8 

 year"M, t^ (0 to 4 years), and K (the von Berta- 

 lanffy growth parameter; 0.1 to 0.5 year"M on 

 MSY (maximum sustainable yield), expressing 

 the result as "MSY as a % of Bq," where Bq is the 

 initial or recruited biomass. Higher percentages 

 indicate that more of the original biomass is 

 being taken at MSY. Increasing M by a factor of 

 8 (0.1 to 0.8 year-i) increased MSY/Bq by a factor 

 of about 4 to 8, depending on the particular values 

 of tf. and K. Again, errors in M produced roughly 

 the same relative error in the result; and again 

 the actual effect of any given change in M de- 



Table 2— Sensitivity of estimated maximum yield per recruit ((V/Rj^ax) 'o changes in instantaneous 

 rate of natural mortality (M) and other input conditions. Sensitivity of (>^'W)max ^^d of changes in Ware 

 expressed as percentage difference from nominal responses at nominal (best-guess) M. Symbols 

 are: t^ = age-at-first-capture, F = instantaneous rate of fishing mortality, M = nominal value for 

 M. Frances (1974) used an age-structured simulation model. All other citations used standard yield- 

 per-recruit analyses. 



Input conditions 



% change % change in 



in M (^/'^Jmax Species 



Source 



plaice 



plaice 



Beverton and Holt 1957 



Beverton and Holt 1957 



yellowfin tuna Hennemuth 1961 

 yellowfin tuna Francis 1974 

 yellov\rfin tuna Bartoo and Coan 1979 



31 



